On complete intersections containing a linear subspace
Francesco Bastianelli2019, Geometriae Dedicata
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Abstract
Consider the Fano scheme F k (Y) parameterizing k-dimensional linear subspaces contained in a complete intersection Y ⊂ P m of multi-degree d = (d 1 ,. .. , ds). It is known that, if t := s i=1 d i +k k − (k + 1)(m − k) 0 and Π s i=1 d i > 2, for Y a general complete intersection as above, then F k (Y) has dimension −t. In this paper we consider the case t > 0. Then the locus W d,k of all complete intersections as above containing a k-dimensional linear subspace is irreducible and turns out to have codimension t in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general [Y ] ∈ W d,k the scheme F k (Y) is zero-dimensional of length one. This implies that W d,k is rational.
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